[This is a quick post I wrote while my younger son was at a little math enrichment activity. Sorry it looks like it was written in a hurry and not proof read . . . ]

Earlier today my older son and I played around with Patrick Honner’s Pi Day exercise:

But, extending to 4 dimensions isn’t as easy as it seems. For one thing, the “volume” and “surface area” of a 4 dimensional sphere involve $\latex \pi^2$ not :

“Volume” =

“Surface Area” =

So, we’ll modify Honner’s 3d formula to be = (1/128) (Surface Area^4) / (Volume^3). That’ll give us a value for and then we can compute .

So, I found the “volume” and “surface area” of the 4 dimensional regular Polytopes here: